In an amplitude modulation, a modulating signal having amplitude of $$X$$ V is superimposed with a carrier signal of amplitude $$Y$$ V in first case. Then, in second case, the same modulating signal is superimposed with different carrier signal of amplitude $$2Y$$ V. The ratio of modulation index in the two case respectively wil be:

We need to find the ratio of modulation indices in two cases of amplitude modulation.

Formula: The modulation index in amplitude modulation is defined as:

$$\mu = \frac{A_m}{A_c}$$

where $$A_m$$ is the amplitude of the modulating signal and $$A_c$$ is the amplitude of the carrier signal.

Modulating signal amplitude = $$X$$ V, Carrier signal amplitude = $$Y$$ V

$$\mu_1 = \frac{X}{Y}$$

Same modulating signal amplitude = $$X$$ V, Carrier signal amplitude = $$2Y$$ V

$$\mu_2 = \frac{X}{2Y}$$

$$\frac{\mu_1}{\mu_2} = \frac{X/Y}{X/2Y} = \frac{X}{Y} \times \frac{2Y}{X} = 2$$

Therefore, $$\mu_1 : \mu_2 = 2 : 1$$.

The correct answer is Option C: $$2 : 1$$.